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I was wondering what the difference between $c_n$ and $c_{jn}$ is? In the context of molecular orbital construction, for example, if $c_{jn}$ is the coefficient of the basis functions $\phi_j$ (contribution of the constituting atomic orbitals), what would $c_n$ be?

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    $\begingroup$ $c_n$ is an eigenvector, $c_{jn}$ is a value of an eigenvector $\endgroup$ – Fl.pf. May 30 '17 at 17:08
  • $\begingroup$ Ah, the commonest of sins in mathematical chemistry, elimination of indices without changing the parent symbols... $\endgroup$ – pentavalentcarbon May 30 '17 at 18:21
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    $\begingroup$ Isn't Math.SE a proper place for this question? $\endgroup$ – Ruslan May 30 '17 at 19:34
  • $\begingroup$ @pentavalentcarbon yeah wherever that's from, it's really bad.... $\endgroup$ – Fl.pf. May 31 '17 at 6:06
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    $\begingroup$ @Ruslan Under the strictest rules, this does belong on Math.SE. However, I argue that this is a prime example of fundamental quantum chemistry (with the example given) and one where notation is important. IMO, chemists tend to be sloppy here. $\endgroup$ – pentavalentcarbon May 31 '17 at 13:34

$\mathbf{c}_n$ is a vector with $N$ entries (I switched to boldface to indicate - hopefully slightly more clearly - that it is a vector):

$$\mathbf{c}_n = \begin{pmatrix}c_{1n} \\ c_{2n} \\ \vdots \\ c_{Nn}\end{pmatrix}$$

This effectively means the same thing as writing

$$\psi_n = c_{1n}\phi_1 + c_{2n}\phi_2 + \cdots + c_{Nn}\phi_N = \sum_j c_{jn}\phi_j$$

I'll try to contextualise it. The columns of the eigenvector matrix are simply the eigenvectors themselves, so if (for example) the eigenvectors of an operator $\mathbf{A}$ are

$$\mathbf{c}_1 = \begin{pmatrix}p \\ q\end{pmatrix}; \qquad \mathbf{c}_2 = \begin{pmatrix}r \\ s\end{pmatrix}$$

then the corresponding eigenvector matrix $\mathbf{C}$ is

$$\mathbf{C} = \begin{pmatrix} \mathbf{c}_1 & \mathbf{c}_2 \end{pmatrix} = \begin{pmatrix} p & r \\ q & s \end{pmatrix}$$

[As an addendum: If the eigenvectors $\mathbf{c}_i$ are normalised, then you will find that

$$\mathbf{C}^{-1}\mathbf{A}\mathbf{C} = \begin{pmatrix} \lambda_1 & 0 \\ 0 & \lambda_2 \end{pmatrix}$$

where $\lambda_i$ is the eigenvalue of $\mathbf{c}_i$, i.e. $\mathbf{Ac}_i = \lambda_i\mathbf{c}_i$.]

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